MATH 446 / OR 481
SAUER
SPRING 2010
Review Problems
27 April 2010
1. Describe the most efficient method you can find (in the sense of the minimum number of
multiplications needed) for evaluating the polynomial
P
(
x
) =
a
3
x
3
+
a
7
x
7
+
a
11
x
11
+
a
15
x
15
+
a
19
x
19
+
a
23
x
23
.
How many multiplications are needed for each input value of
x
?
2. Find the IEEE doubleprecision machine representation of the decimal numbers (a)
15
(b)
1
/
8
(c)
0
.
8
(d)
0
.
7
3.
(a) Convert the binary number
11
.
1
001
to base ten.
(b) Convert
31
/
6
to binary.
(c) If
31
/
6
is to be stored as a double precision number using IEEE rounding to nearest,
find the exponent, and list the four rightmost bits in the stored number.
4. Calculate (a)
(2 + (2

51
+ 2

52
))

2
and (b)
(2 + (2

51
+ 2

52
+ 2

53
))

2
and express
the answers in terms of machine epsilon.
5. Calculate both roots of
x
2
+ 2
60
x
= 1
to 3 correct decimal digits.
6. (a) Solve
36
x
2
+ 36
x
= 7
for the degree 1 term and set up the resulting fixed point iteration.
(b) Show that
1
/
6
and

7
/
6
are fixed points. (c) Which of the fixed points are locally con
vergent? (d) If FPI is run with initial guess
x
= 1
, what will happen? (e) Same as (d), but for
initial guess
x
= 2
.
7. (a) Show that
x
= 4
/
3
and
x
= 5
/
3
are fixed points of the equation
x
= (
x

1)
2
+ 11
/
9
.
(b) Which fixed point will attract initial guesses under FPI? (c) If FPI is run with initial guess
x
= 1
, what will happen? (d) Same as (c), but for initial guess
x
= 2
.
8. Develop a method for computing the fifth root of a number
a
to several correct digits. It should
require only elementary operations like multiplications, divisions, additions, etc. Discuss the
convergence properties of your algorithm.
9. Assume that Newton’s method is applied to find the roots of
f
(
x
) =
x
4
+ 2
x
3

2
x

1
, and
assume that after 4 steps you are within
e
4
= 0
.
0001
of the root. Estimate the total number of
steps required to calculate the following roots within 50 correct decimal places: (a)
r
=

1
(b)
r
= 1
.
10. For
f
(
x
) =
x
4

(7
/
2)
x
3
+ (15
/
4)
x
2

(13
/
8)
x
+ 1
/
4
, does Newton’s method converge
faster or slower than the bisection method to
x
= 1
/
2
? What about to
x
= 2
?
11. Let
f
(
x
) =
x
3
+ 2
x

3
.
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 Spring '08
 Staff
 Numerical Analysis, Multiplication, Polynomial interpolation, point iteration, Sauer

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